Find the relations between A, B and C when $[(A\cap B)\cup C]-A=(A\cap B)-C$

So we can write it as: $[(A\cup C)\cap(B\cup C)]-A=(A\cap B)-C$. Here comes the problem, though. Can I just assume that $[(A\cup C)\cap(B\cup C)]=(A\cap B)$? If I can, I can go on from there with: $C\subseteq A, C\subseteq B$ and eventually for the whole thing to hold $A=C$ which then shows that $A\subseteq B$. Can I do this that way, however? I feel that I'm assuming too much by simply comparing $[(A\cup C)\cap(B\cup C)]$ and $(A\cap B)$ to each other.

1 Answer
1

Let me name the two sets:
$$
P=[(A\cap B)\cup C]-A\qquad Q=(A\cap B)-C
$$
Your task is to determine the conditions on $A, B, C$ which will ensure that $P=Q$.

Helpful result 1: $P=C-A$.

A Venn diagram will make this clear, or you can show it by using set equivalences.

Helpful result 2: $P\cap Q = \varnothing$.

Again, use a Venn diagram or set equivalences. As a consequence we must conclude that both $P\text{ and }Q$ are empty, since the only way two sets can be equal but have no elements in common is for them both to be empty.

So we have two conditions:
$$
\begin{align}
P &= [(A\cap B)\cup C]-A=\varnothing\\
Q &= (A\cap B)-C=\varnothing
\end{align}
$$
Let's look at the first condition, $P=C-A=\varnothing$. This is true if and only if
$C\subseteq A$.

Now apply this condition to $Q$. As above, $(A\cap B)-C=\varnothing$ if and only if $A\cap B\subseteq C$, so our condition is that

$P\text{ and }Q$ are equal if and only if $A\cap B\subseteq C\subseteq A$.